economic geography and endogenous determination of transportation...
TRANSCRIPT
Economic Geography and Endogenous Determination of
Transportation Technology
Takaaki Takahashi
The University of Tokyo
CORE
January 2005
Abstract
This paper studies the interdependence of economic geography and transportation technol-
ogy. A two-region model is used to obtain the conditions for the modern transportation
technology to be adopted in an economy. In particular, the impact of economic geography
upon the adoption of the modern technology is examined. Furthermore, I discuss what com-
bination of economic geography (symmetric or core-periphery pattern) and transportation
technology (traditional or modern technology) is to be realized in an economy.
Keywords: core-periphery pattern; lock-in effect; modern transportation technology; sym-
metric pattern; traditional transportation technology; transportation cost
JEL Classification Numbers: F11 (Neoclassical Models of Trade), R13 (General Equi-
librium and Welfare Economic Analysis of Regional Economies)
I would like to thank Kristian Behrens and Tomoya Mori for stimulating discussions and comments. Iam greatly benefited from the comments by Masahisa Fujita, Takatoshi Tabuchi and Jacques Thisse. I alsoappreciate comments from seminar participants at various institutions. Parts of the research was conductedwhile I visited the CERAS, Ecole Nationale des Ponts et Chausees, Paris, France, whose hospitality is greatlyappreciated. This research is supported by the Grant in Aid for Research (No. 13851002) by Ministry ofEducation, Science and Culture in Japan.
1 Introduction
Since the seminal work by Krugman (1991), a number of theoretical and empirical studies
have been conducted to strengthen our understanding of location of economic activities. One
of the basic messages of the “new economic geography” is that transportation cost matters:
it affects the decision making of each producer and consumer and, as a result, determines
the emerging geographical patterns.1 Krugman (1991), for instance, constructs a simple
two-region model to show that low transportation cost tends to cause an agglomeration of
economic activities while high cost a dispersion. This result recurs in various contexts in
most chapters of Fujita, Krugman and Venables (1999) and Fujita and Thisse (2002), which
examine such diverse topics as regional development, urban systems and international trade.2
Surprisingly, however, few researchers have discussed the opposite causality, that is, the
causality from the economic geography to the transportation cost. Casual observations
suggest that the transportation technology adopted in an economy depends on the location
of economic activities within it. A good example may be provided by cities. In cities like
Los Angeles where the economic activities are highly dispersed over a broad range of space,
a considerable portion of transportation is made by automobiles. In contrast, cities like
Paris where the activities are historically concentrated in a narrow district usually see the
development of mass transportation systems such as subways and trams, probably because
the concentration generates enough demand for their services to cover the cost of their
construction. Thus, the economic geography affects the adopted technology and, as a result,
the transportation cost.
Studying such a relationship between economic geography and transportation technology
is a matter of great importance for several reasons.
First, we often observe that some countries succeed in adopting a “modern” technology
like a railroad whereas the others, although similar in most aspects, especially in their
levels of economic development, fail to do so, sticking to a “traditional” technology like
a motorcycle. This puzzling observation may be well explained by the difference in the
economic geography of each country.
Second, the adopted technology determines not only the current level of welfare but also
the future path of economic development and consequently its future level. In this regard,
studying the adoption of technology is the more important.
Third, the topic is closely related to the problem of coordination in economic development1For recent overviews of the field, see Ottaviano and Puga (1998), Fujita and Thisse (2000) and Neary
(2001), among others.2While most research including Fujita, Krugman and Venables (1999) deal with a general level of the
transportation cost, Behrens (2004) pays attention to its scheme with respect to the distance of shipping.
He explores the effects of the difference in the scheme upon the economic geography.
1
and may have some important policy implications. Suppose that, for example, the agglom-
eration of economic activities yields a sufficient amount of demand for the transportation
service based on a superior modern technology, whereas dispersion does not. Then, that
technology will be adopted in the economy characterized by the agglomeration but not in
that characterized by the dispersion. In other words, the coordination among producers to
locate themselves in the same region is necessary for the adoption of the modern technology.
In this case, consequently, economic policies should aim at promoting such a coordination.
Thus, this study offers another version of the famous ‘Big Push’ story by Murphy, Shleifer
and Vishny (1989).
Fourth, combining both directions of the causality between the economic geography
and the transportation technology, we can complete the picture of a circular causation or
positive feedback mechanism. Suppose, as in the earlier discussion, that agglomeration
rather than dispersion is associated with the superior transportation technology. Then, if
a sufficient fraction of economic activities happens to be concentrated in one region, the
superior technology is adopted. As a result of its adoption, the transportation cost will
decline, which will, in turn, strengthen the tendency toward the agglomeration according to
the mechanism described in the standard literature of new economic geography.
Fifth and finally, in order to study the adoption of transportation technology, it is in-
evitable to explicitly incorporate into the analysis a transportation sector, which has been
usually abstracted away in the literature.3 Nonetheless, it is likely that the sector is char-
acterized by some form of increasing returns, e.g., the increasing returns to scale as Neary
(2001) points out and the increasing returns to density as Mori and Nishikimi (2000) dis-
cuss. Since the increasing returns is another corner stone of the new economic geography,
the omission of the transportation sector is not too innocuous.
This paper is one of the first attempts at studying the interdependence of emerging eco-
nomic geography and adopted transportation technology. For that purpose, I construct a
two-region model with two goods, an intermediate and a final good, in spirit of the new
economic geography. It takes some cost to ship the final good from one region to the other
although no cost is necessary to ship the intermediate. There are mobile entrepreneurs pro-
ducing the final good and immobile workers producing the intermediate. For the geography,
I focus on two special distribution patterns of entrepreneurs, namely, a symmetric pattern,
at which they are distributed equally in the two regions, and a core-periphery pattern, at
which they are concentrated in one region. Furthermore, I extend the conventional frame-
work so as to incorporate two sorts of inter-regional transportation technologies into the
analysis, “traditional” and “modern” technologies. The former, which is used by consumers3In the literature, they often assume that the transportation cost takes an ‘iceberg’ form, which enables
us to discuss the cost without explicitly modeling the transportation sector.
2
when they themselves carry the final good on foot, is characterized by constant returns.
The latter is applied by the transportation sector that constructs a transportation system
and charges a fee for its use. Since the construction of the system involves a fixed cost, it
is subject to increasing returns. For the sake of simplicity, I concentrate on the situation in
which the level of the user fee is determined so that the revenue from it exactly covers the
fixed cost. The modern technology is adopted in an economy when the level of its user fee
is sufficiently low and, as a result, the associated transportation cost becomes low enough
to beat the traditional technology.
The first question to ask is as follows: in which distribution pattern is the modern
transportation technology more likely to be adopted? Given the transportation cost for the
traditional technology, the lower the transportation cost for the modern technology is, the
more likely the latter technology is to be adopted. To answer the question, therefore, it
suffices to find out the distribution pattern at which the transportation cost is lower.@Now,
the level of the transportation cost is determined by that of the user fee: the higher the
user fee is, the higher the transportation cost is. Furthermore, the user fee that exactly
covers the fixed cost declines with the demand for transportation service, which is positively
related to the volume of inter-regional trade. Hence, the transportation cost becomes lower
at the distribution pattern with a larger volume of inter-regional trade. Thus, we cay say
that the modern technology is more likely to be adopted at the distribution pattern with
a larger volume of inter-regional trade. Here, it is important to note that the volume of
trade itself depends on the transportation cost: the transportation cost for the modern
technology and the volume of trade are inter-related and must be solved simultaneously.
Paying attention to this inter-dependence, I give the answer to the question of which pattern
yields a larger volume of trade. The answer turns out to depend on the values of parameters,
i.e., the elasticity of substitution in consumers’ preference, the cost to construct a modern
transportation system and the efficiency of the modern technology.
Next, I ask what combination of economic geography (symmetric or core-periphery pat-
tern) and transportation technology (traditional or modern technology) is to be realized
in an economy. Here, in addition to the question of which technology is adopted for a
given geography, one needs to answer the question of which geography emerges for a given
technology. As an answer to the latter question, I consider a stable long-run equilibrium
distribution pattern. It turns out that the realized combination depends on the values of
parameters in a highly complicated way. Thus, I would rather pay attention to a couple of
interesting cases than enumerate all the cases in more or less great detail.
Some of them are the cases exhibiting a poverty trap, in which the economy fails to adopt
the modern technology despite the fact that it could do so if the economic geography were
3
different. More specifically, I show that there exists a set of parameter values for which the
following holds: When the economy is characterized by the symmetric pattern, on the one
hand, the modern technology, being prohibitive, cannot be introduced into it and the tradi-
tional technology is adopted. Given the traditional technology, furthermore, the symmetric
pattern is a stable long-run equilibrium pattern. On the other hand, when it is characterized
by the core-periphery pattern, the modern technology is affordable enough to be adopted in
the economy. Given the modern technology, the core-periphery pattern is a stable long-run
equilibrium pattern. For such a set of parameter values, the economy with the symmetric
pattern is locked in to the inferior state, namely, that with the traditional technology, that
is to say, it is stuck into the poverty trap. Studying this lock-in effect is important because
of not only its theoretical significance but also its far-reaching implications. For one thing,
it gives us a justification for policy intervention. What is more, it provides us with an expla-
nation of the fact which I have mentioned earlier, i.e., the fact that some countries succeed
in adopting the modern technology while the others cannot even if their levels of economic
development are not much different.
Finally, I discuss how the adopted transportation technology and the realized economic
geography change in response to a gradual change in economic environment. This would be
helpful to understand some aspects of economic development.
The paper is organized as follows. In the next section, I present a basic model. Section 3
gives the transportation costs for the modern technology when the economy is characterized
by the symmetric pattern and by the core-periphery pattern, respectively. In the subsequent
section, I compare these costs to see which pattern the modern technology is more likely
to be adopted at. In Section 5, I define the stable long-run equilibrium pattern. Then,
in the next section, I ask what combination of transportation technology and economic
geography is realized in an economy. The possibility of the lock-in effect and the change in
the combination brought about by a gradual change in an exogenous factor associated with
economic development are discussed. Finally, Section 7 concludes.
2 Model
As a basic framework, I use a model presented by Mori and Turrini (2000), extending it so
as to allow the possibility that there exist different types of transportation technologies and
a transportation sector.
4
2.1 Basic Framework
There are two regions, denoted by 1 and 2; and two goods, an intermediate and a final
good. Total labor force consists of workers and entrepreneurs. The workers are not endowed
with the skill necessary to produce the final good and thus engage in the production of the
intermediate. There are 2 units of workers in the economy, who cannot migrate between the
regions. They are distributed equally in the two regions: 1 unit live in each region. The
entrepreneurs are, on the other hand, endowed with the skill so that they can engage in the
production of the final good. They can freely move between the two regions. Their number
is fixed at n, of which λn live in region 1 and (1− λ)n live in region 2 (λ ∈ [0, 1]).
The intermediate is a homogeneous product and produced in a competitive sector. Each
worker produces 1 unit of the intermediate, whose price, therefore, equals his wage rate in
each region. The cost to ship the intermediate from one region to the other is assumed to
be 0 so that the prices of the intermediate and, consequently, the wage rates of workers are
equalized in the two regions. I consider the wage rate a numeraire.
On the contrary, the final good is a differentiated product and produced in a monopolis-
tically competitive sector. Each entrepreneur owns a firm which produces 1 variant in the
region of her residence. Thus, there are n variants in the economy of which λn are produced
in region 1 while (1− λ)n are produced in region 2. Taking a unit appropriately, I suppose
that each firm produces 1 unit of a variant from 1 unit of the intermediate using the skill of
the respective entrepreneur. All the revenue left after the payment for intermediate is taken
by the entrepreneur. In other words, the profit of a firm is equal to 0:
pq − 1 · q − w = 0, (1)
where p and q denote a price and an amount of a variant produced, and w a wage rate
received by an entrepreneur. This setting must be familiar in the analytically solvable core-
periphery model (Forslid and Ottaviano (2003)).
The workers and entrepreneurs have the same preference represented by a utility function,
U = [∫ n
0x(k)ρ dk]1/ρ, where ρ ∈ (0, 1), and x(k) denotes the amount of the kth variant. Let
us denote the amount of money that a consumer must pay to consume 1 unit of the kth
variant by p(k). Then, the standard analysis tells us that a consumer, whose income is equal
to y, consumes the following amount of the kth variant:
x(k) = yp(k)−σPσ−1 (2)
where σ ≡ 1/(1− ρ) > 1 is the elasticity of substitution and
P =[∫ n
0
p(l)1−σ dl
]1/(1−σ)
(3)
5
is the price index.
The entrepreneurs sell their variants at the price that maximizes their own wages. Since
not only they are subject to the same technological condition but also all the variants enter
the utility function in a symmetric manner, they charge the same price as long as they are
located in the same region. Thus, I denote the prices of a variant produced in respective
regions by p1 and p2.
Next, I introduce a transportation cost. In order to consume 1 unit of any variant in
region j, it is necessary for consumers to buy tij ≥ 1 units of that variant if it is produced
in region i. If i = j that is, the regions of production and consumption are the same,
tij = 1: no transportation cost is necessary. However, if i 6= j, tij > 1: they must pay some
transportation cost. I assume that the cost does not depend on the direction of shipment,
that is, t12 = t21 = t > 1. To sum up, we have
tij =
1 if i = j, and
t otherwise.
The transportation cost, tij − 1 may be interpreted either as the amount of the good that
melts away during the shipment, as in the standard literature, or as the amount that is
received by a transportation sector. In either case, a consumer in region j who consumes 1
unit of the kth variant produced in region i must pay p(k) = tijpi.
Let Xjj and Xij be the total demands, inclusive of transportation cost, in region j for
a variant produced in that region and in the other region, respectively. By definition, it
follows from (2) and (3) that
Xjj = Yjp−σj Pσ−1
j ( j = 1, 2 ) (4)
and that
Xij = Yjt1−σp−σ
i Pσ−1j ( j = 1, 2 ) (5)
with i 6= j, where Yj is an aggregate income and
Pj =[n
λ(t1jp1)1−σ + (1− λ)(t2jp2)1−σ
]1/(1−σ)( j = 1, 2 ) (6)
is a price index in region j. Since the amount of each variant produced in region i, qi, must
be equal to its demand, we have
qi = Xii + Xij ( i = 1, 2 ) (7)
with j 6= i.
An entrepreneur in region i maximizes her wage given by wi = piqi − 1 · qi ( i = 1, 2 )
(see (1)), setting the price at
p1 = p2 =σ
σ − 1. (8)
6
The wage received by an entrepreneur is reduced to
wi =1
σ − 1qi. ( i = 1, 2 ) (9)
Moreover, the aggregate income consists of workers’ and entrepreneurs’ earnings. It is,
therefore, given by
Y1 = 1 + λnw1
Y2 = 1 + (1− λ)nw2.(10)
This completes a description of a basic framework: a system of equations (4) to (10) deter-
mines a short-run equilibrium, in which entrepreneurs’ distribution, λ, is given.
I pay my attention to two special distribution patterns of entrepreneurs; a symmetric
pattern, in which they are distributed equally in the two regions, namely, λ = 1/2, and a
core-periphery pattern, in which all of them are concentrated in one region, namely, λ = 0
or λ = 1. Those two patterns are denoted by S and C, respectively.
2.2 Transportation Technology and Transportation Sector
In the economy, there are two types of transportation technology. The first is a traditional
technology, denoted by T . For this technology, a fixed portion of the final good melts away
in course of inter-regional shipment. Specifically, 1 unit of the good arrives at a destination
when tT > 1 units are dispatched, that is to say, t = tT . For this technology, consumers
carry the good by themselves “on foot”.
Then, there is a modern technology, denoted by M . Using it, consumers can receive 1
unit of the final good by dispatching merely γ units instead of tT units (γ ∈ (1, tT )) at the
opposite region. Here, γ is inversely related to “efficiency” of the modern technology.
The transportation service based on the modern technology is provided by a transporta-
tion sector, which may be private or public. It uses F units of the final good in order to
construct, maintain and operate a system of transportation such as a highway network and
a high-speed train system. For the sake of simplicity, I assume that the sector provides the
service with 0 marginal cost. Furthermore, the sector takes from users of its system u > 0
units of the final good per every unit that is shipped out from the origin of the shipment.4
Thus, u represents a “user fee” in real terms per unit shipment. The point is that, in order to
receive 1 unit of the final good at one region, one must dispatch γ/(1−u) units at the other
region: out of γ/(1 − u) units, the transportation sector takes portion u, leaving portion
1− u or γ units, which melts away but 1 unit. Thus, when the modern technology is used,
we have t = γ/(1− u).
4In order to keep the analysis tractable, I exclude the possibility that the transportation sector gives a
subsidy for the use of its system.
7
The profit of the transportation sector (in real terms) is given by π ≡ uD − F . Here, D
is the total amount of cross-border demand for the final good or the volume of inter-regional
trade, measured at the origin of the shipment. It is equal to the sum of nλX12 (the amount
of the good produced in region 1 for the consumption in region 2) and n(1 − λ)X21 (the
amount of the good produced in region 2 for the consumption in region 1):
D = n [λX12 + (1− λ)X21] .5 (11)
In this paper, I focus on the special case where the profit is equal to 0, that is, the case
where the user fee is determined by
uD − F = 0, (12)
whenever it is in operation. The purpose is to keep the analysis tractable: taking into ac-
count the possibility that the sector earns positive profits makes the analysis much more
complicated. Since the 0-profit case, as will be shown, becomes an important benchmark
for the more general cases with positive profits, this simplification is not too harmful as
a first step. Furthermore, it should be noted that the 0-profit case can be interpreted in
several ways. First, a government provides the modern transportation service with average
cost pricing to balance its budget. Second, a government regulates a privately-owned trans-
portation sector so that no profit is raised. Third and finally, as a result of (potential) free
entry, a privately-owned transportation sector is forced to charge the 0-profit user fee.
Now, what conditions are necessary for the adoption of a particular type of transporta-
tion technology? First, the technology must be available to the consumers. The modern
technology is available if there exists a 0-profit user fee. If it does not exist, the transporta-
tion sector cannot earn a non-negative profit, as will be shown later, and, as a result, stays
out of business. The traditional technology is always available. Second, the technology must
be feasible. As has been mentioned, the transportation sector takes away some portion of
the final goods from those that a consumer has passed to it. However, it is not feasible to
take away more than s/he has passed. Therefore, the modern technology is feasible if u ≤ 1,
orγ
1− u≥ 0. (13)
We can consider the traditional technology to be always feasible because no user fee is taken
(u is regarded as 0). Third and finally, when both the technologies are available and feasible,
the adopted technology must be actually used by consumers who choose the less expensive5Using (4), (5), (7)-(9), we can rewrite (10) in terms of Y1, Y2, P1 and P2. Then, solving for Y1 and Y2
yields YiPσ−1i = Zi/(A1A2 − B1B2) where Z1 ≡ A2 + B1, Z2 ≡ A1 + B2, Ai ≡ P 1−σ
i − ρσ−1nλi/σ,
and Bi ≡ ρσ−1t1−σnλi/σ (λ1 ≡ λ and λ2 ≡ 1 − λ) (i = 1, 2). Furthermore, this gives us D =
ρσt1−σn [(1− λ)Z1 + λZ2] /(A1A2 −B1B2) .
8
mode of transportation. In other words, the transportation cost for the adopted technology
must be no higher than that for the other technology: the adopted technology must undercut
the other technology. Here, in order to avoid an unnecessary complication, I assume that it
is the modern technology that is used when the tie occurs.6 Then, if
γ
1− u≤ tT , (14)
the modern technology undercuts the traditional technology; and otherwise, the opposite
holds.
Now, using these concepts, we can formally discuss the conditions for the adoption of a
transportation technology.
Definition 1. (adoption of a transportation technology) Given a distribution pattern,
a transportation technology i ∈ T,M is said to be adopted in the economy if one of the
following two sets of conditions, i) and ii), is met for j ∈ T,M with j 6= i:
i) technology i is available and feasible, while technology j is unavailable and/or infeasible,
ii) technology i is available and feasible, and undercuts technology j, while technology j is
available and feasible.
The following lemma, whose proof is relegated to Appendix, immediately follows.
Lemma 1. Given a distribution pattern, the modern transportation technology is adopted
in the economy and t = γ/(1−u), if it is available and feasible, and undercuts the traditional
technology. Otherwise, the traditional technology is adopted and t = tT .
Here, we can see that the 0-profit case becomes a benchmark for more general cases with
positive profits. Suppose that the modern technology is adopted and that the transportation
sector earns a positive profit. Then, the technology is available and feasible, and undercuts
the traditional technology. Now, it will be shown later that the 0-profit user fee is lower than
any fee associated with positive profits. Therefore, the modern technology is still available
and feasible, and undercuts the traditional one when the sector charges the 0-profit fee.
That is, whenever the modern technology is adopted for a user fee associated with a positive
profit, it can be adopted, too, for the 0-profit fee. Taking contraposition, if the modern
technology is not adopted for the 0-profit fee, consequently, it is not adopted, either, for any
fee associated with positive profits.
Closing this section, I add one qualification about the values of parameters. In the
following analysis, it will be shown that if F > ρ, the modern transportation technology,
failing to be feasible, is not adopted when the economy is characterized by the core-periphery6This assumption is made only for the sake of simplicity and clarity. It is possible to assume otherwise
and obtain similar results.
9
pattern. In order to focus on interesting cases, therefore, I preclude such a case, assuming
that the fixed cost is sufficiently small.
Assumption 1. F ≤ ρ.
3 Transportation Costs Associated with the Modern
Transportation Technology
In this section, supposing that the modern technology is adopted in the economy, I derive
the transportation costs associated with it, given the distribution pattern of entrepreneurs.
In doing so, I also explore the three conditions for its adoption.
3.1 Transportation Cost at the Symmetric Pattern
In the first place, let us assume the symmetric distribution of entrepreneurs. When the
modern technology is adopted, the measure of inter-regional transportation cost, t, becomes
equal to tS ≡ γ/(1− uS), where uS is a user fee at the symmetric pattern.
According to (11), the amount of the cross-border demand is reduced to
DS =2
1 + tσ−1S
. (15)
It depends on σ and tS . First, ∂DS/∂σ < 0: the cross-border demand decreases with
σ, other things being equal. This is explained as follows. At the symmetric pattern, the
cross-border demand originates from not only the workers but also the entrepreneurs. As
the elasticity of substitution rises, monopoly power of the entrepreneurs is abated. As a
result, the price of the final good declines, which lowers their nominal income (a nominal
income effect), on the one hand, and the price index (a price index effect), on the other
hand. Since the former effect, it turns out, dominates the latter, their demand for the final
good shrinks. Although the decline in the price expands the demand from the workers at
the same time, it is more than offset by the shrinkage in the entrepreneurs’ demand. This
results in the reduction of the cross-border demand. Second, ∂DS/∂tS < 0: the cross-border
demand decreases with tS , ceteris paribus. This is because the rise in the transportation
cost induces the substitution of the variants produced in the home region for those produced
in the foreign region.
Substituting DS into the profit function yields
π =Ω(tS)
tS(1 + tσ−1S )
(16)
10
where Ω(tS) ≡ tS(2− F )− FtσS − 2γ. Therefore, the transportation sector earns 0 profit if
tS solves
Ω(tS) = 0. (17)
By definition, the modern technology is available when (17) has a solution. Furthermore,
note that Ω(tS) approaches the negative infinity as tS goes to the positive infinity. By
continuity, therefore, when there exists no solution to (17), Ω(tS) < 0 for any tS , that is,
the transportation sector ends up with a negative profit.
Whether (17) has a solution or not depends on the values of parameters. Note that the
solution exists if and only if
maxtS
Ω(tS) ≥ 0. (18)
Since Ω(tS) is concave, it has a unique maximum, reached at t0S ≡ [(2− F )/σF ]1/(σ−1)> 0,
where the inequality is assured by Assumption 1. Two observations follow. First, because
d[max
tS
Ω(tS)]
dσ= − t0S(2− F )
σ(σ − 1)ln
2− F
σF,
maxtS
Ω(tS) decreases with σ if 1 < σ < (2− F )/F , and increases if σ > (2− F )/F . Second,
maxtS
Ω(tS) approaches the positive infinity as σ goes to 1 from above, and −2γ as it goes
to the positive infinity. These observations imply that there exists σ ∈ (1, (2− F )/F ) such
that max
tS
Ω(tS) > 0 for σ < σ,
maxtS
Ω(tS) = 0 for σ = σ, and
maxtS
Ω(tS) < 0 for σ > σ.
Because maxtS
Ω(tS) = Ω(t0S), σ is a solution to
Ω([(2− F )/σF ]1/(σ−1)
)= 0. (19)
Hence, (18) is satisfied and (17) has a solution if and only if σ ≤ σ, which is the condition
for the modern technology to be available.
A figure might be helpful to understand this finding. In Fig. 1, the horizontal axis
measures σ while the vertical axis measures tS . The tS(σ) curve represents the solution to
(17) as a function of σ for given values of γ and F ( γ = 1.1 and F = 0.1). It has a turning
point at σ = σ, where the height is equal to t0S |σ=σ (see (19)).
@Insert Fig. 1 around here
It may be worth noting that this availability condition can be rewritten in terms of the
other parameters. Let us begin with parameter F . Notice that maxtS
Ω(tS) decreases with F ,
approaches the positive infinity as F goes to 0, and approaches a negative number as it goes
11
to 1. Therefore, (18) is satisfied and a solution to (17) exists, if and only if F is no greater
than a critical value, F ∈ (0, 1), which is a solution to maxtS
Ω(tS) = 0. We can obtain a
similar result for γ: a solution to (17) exists if and only if γ is no greater than a critical value,
γ > 0, which is a solution to maxtS
Ω(tS) = 0.7 It turns out that γ = ρ [(2− F )σ/σF ]1
σ−1 /2.
Having said that (17) has a solution if σ ≤ σ, I now derive several properties of the
solution. First of all, (17) has at most two solutions since Ω(·) is strictly concave. I denote
the solutions as tS and tS with tS ≤ tS . By construction, we have
tS ≤ t0S ≤ tS . (20)
The two solutions, tS and tS , are represented by the heights of the increasing part and
decreasing part of the tS(σ) curve in Fig. 1, respectively. It is worth noting that tS , a low
price, is associated with high volume of inter-regional trade while tS , a high price, is with low
volume of inter-regional trade. Second, note that Ω(tS) > 0 for any tS ∈ (tS , tS), and that
Ω(tS) < 0 for any tS < tS and for any tS > tS . This immediately follows from the concavity
of Ω(·). Third, all the solutions are greater than γ, that is, tS > γ. To see this, notice that
Ω(t0S) > 0 implies that t0S > 2γ/ [ρ(2− F )] > γ. However, Ω(γ) = −γF(1 + γσ−1
)< 0,
which implies γ < tS or γ > tS . Consequently, tS > γ. Fourth and finally, the elasticity of
the cross-border demand with respect to the user fee is not greater than 1 at tS . This is
because
−d lnDS
d lnuS=
(σ − 1)tσ−1S
1 + tσ−1S
· tS − γ
γ=
(σ − 1)Ftσ−1S
2− F (1 + tσ−1S )
,
where (15) and (17) are used: applying (17) and (20), we can easily verify that d lnDS/d lnuS ∈
(0, 1] at tS .
Of the two solutions, I will limit my attention to tS in the subsequent analyses. There
are two reasons. First, tS is associated with a higher level of social welfare than tS is. This
is because the former is closer to the marginal cost in the provision of the transportation
service, which is 0, than the latter. Viewing it from a different angle, moreover, one can
recall that tS involves a low price and a high volume of inter-regional trade while tS a high
price and a low volume of inter-regional trade. Since the profit is 0 for both situations, it is
obvious that the former is associated with a higher welfare. Therefore, a government will, if
any, regulate tS at tS rather than tS . Second, it is not likely that tS is implemented in the
real world. As long as tS > t0S , Ω(tS) is decreasing around tS . Then, from (16), it is clear
that the profit decreases with tS : the elasticity of the cross-border demand is so high that
the sector could earn a higher profit by cutting its user fee. This is not the situation that
we usually observe.8
7Note that maxtS
Ω(tS) decreases with γ, approaches a positive number as γ goes to 0, and approaches the
negative infinity as it goes to the positive infinity.8In almost all cases, governments set an upper limit but not a lower limit of the user fee when they
12
It is important to note that tS > 0 since tS > γ. Therefore, (13) is always satisfied:
the modern technology is always feasible for the symmetric distribution. Furthermore, the
undercutting condition is written as tS ≤ tT .
Lastly, some comparative statics analyses would be helpful to understand the structure
of the model. Let us define KS ≡
σF[(
t0S)σ−1 − tσ−1
S
]−1
> 0.
First, we have ∂tS/∂σ = KSFtσS ln tS > 0: the transportation cost increases with the
elasticity of substitution. This is explained as follows. Suppose that the elasticity rises.
As is discussed earlier, this reduces the cross-border demand through the decline in the
price of the final good (∂DS/∂σ < 0). If the user fee were kept constant, therefore, the
fixed cost would exceed the revenue of the transportation sector. In order to restore the
equality between them, consequently, it is necessary to raise the fee because the elasticity
of the cross-border demand is not greater than 1, which has been shown earlier. Thus, the
break-even user fee and, therefore, the associated transportation cost must rise.
Second, ∂tS/∂F = 2KS(tS−γ)F−1 > 0: the transportation cost increases with the fixed
cost. In order to keep the revenue equal to the fixed cost, the rise in the fixed cost must
be offset by the rise in the revenue, which is attained through the rise in the user fee and,
therefore, that in the transportation cost.
Finally, ∂tS/∂γ = 2KS > 0: the transportation cost is higher for a less efficient mod-
ern technology (corresponding to higher γ). The reason is twofold. First, a less efficient
technology would be associated with a higher transportation cost by definition even if the
transportation sector charged the same fee. Second, the transportation sector indeed charges
a higher fee for a less efficient technology. This is explained as follows. Other things being
equal, a less efficient technology, which is associated with a higher transportation cost, re-
sults in a smaller cross-border demand (∂DS/∂tS < 0) and, therefore, a lower revenue. To
make the revenue even with the fixed cost, the sector needs to charge a higher user fee.
To sum up, given the symmetric distribution pattern, the level of t associated with
the modern technology is given by tS . The technology is available if σ ≤ σ; it is always
feasible; and it undercuts the traditional technology if tS ≤ tT . The following proposition
immediately follows from Lemma 1.
Proposition 1. When the economy is characterized by the symmetric pattern, the modern
transportation technology is adopted and t = tS, if and only if both σ ≤ σ and tS ≤ tT are
satisfied.
regulate transportation sectors. This fact would indicate that the profit of the transportation sector rather
increases with the transportation cost it charges.
13
3.2 Transportation Cost at the Core-Periphery Pattern
Next, let us turn our attention to the economy characterized by the core-periphery pattern.
The user fee is now denoted by uC , for which the measure of transportation cost, t, becomes
tC ≡ γ/(1− uC).
According to (11), the cross-border demand is given as
DC = ρ. (21)
Note that ∂DC/∂σ > 0: the cross-border demand increases with σ, which makes a sharp
contrast to the previous case. This is because, at the core-periphery pattern, the entire
cross-border demand comes from the workers, which implies that any change in the nominal
income of entrepreneurs has no effect on it. Hence, as the elasticity of substitution rises
and the price of the final good declines as a result, the cross-border demand increases
through the price index effect for the workers. Furthermore, ∂DC/∂tC = 0: the level of
the transportation cost does not affect the cross-border demand. Since the workers in the
periphery have no alternative but to buy the final good from the core, the change in the
transportation cost induces no substitution of the variants produced in the home region for
those produced in the foreign region.9
Now, (12) and (21) imply that the break-even level of tC is given by
tC = KCγ(σ − 1), (22)
where KC ≡ (σ − 1 − σF )−1. Thus, there always exists a user fee that yields 0 profit: the
modern technology is always available. Furthermore, (22) explains why we need Assumption
1 for the technology to be feasible. Finally, the modern technology undercuts the traditional
one if KCγ(σ − 1) ≤ tT .
Three additional comments are in order. First, the elasticity of the cross-border demand
with respect to the user fee is 0. Second, tC is represented by the tC(σ) curve in Fig. 1.
Finally, ∂tC/∂σ = KCtCF (σ− 1)−1 < 0, ∂tC/∂F = KCσtC > 0, and ∂tC/∂γ = tCγ−1 > 0.
We can explain these results by the same logic as in the case of the symmetric pattern, except
for two respects. First, as has been discussed earlier, the rise in the elasticity of substitution
does not reduce but does expand the cross-border demand (∂DC/∂σ > 0). Therefore, the
sign of ∂tC/∂σ becomes opposite to that of ∂tS/∂σ. Second, although tC is higher when γ
is higher, the break-even user fee is independent of γ. To see this, suppose that the economy
is now provided with a less efficient technology. Even if the break-even user fee is kept
constant, tC will rise. However, because the amount of the cross-border demand does not9Recall that the cross-border demand has been measured by the amount dispatched at the origin of the
shipment but not the amount that reaches the destination.
14
depend on the transportation cost ( ∂DC/∂tC = 0), the break-even user fee does not need to
change. At the symmetric pattern, on the contrary, the amount of the cross-border demand
decreases with the transportation cost, and, consequently, the break-even user fee needs to
rise.
To sum up, for the core-periphery pattern, the modern technology is always available
and feasible (as long as Assumption 1 holds); and undercuts the traditional technology if
KCγ(σ − 1) ≤ tT . Using Lemma 1, we have established the following proposition.
Proposition 2. When the economy is characterized by the core-periphery pattern, the mod-
ern technology is adopted and t = tC = KCγ(σ − 1), if and only if KCγ(σ − 1) ≤ tT .
4 Which Distribution Pattern Favors the Modern Trans-
portation Technology? A Comparison
Having established the conditions for the adoption of the modern technology, we can now
compare them to ask a question: is the modern technology more likely to be adopted at
the symmetric pattern or at the core-periphery pattern? First of all, recall that the modern
technology is always available at the core-periphery pattern but is not necessarily so at the
symmetric pattern. When it is not available at the latter pattern, the answer is obvious:
the technology is more likely to be adopted at the core-periphery pattern.
When it is available at the symmetric pattern, on the other hand, we need to compare
the conditions on the undercutting, tS ≤ tT for the symmetric pattern and tC ≤ tT for
the core-periphery pattern. We can consider that the modern technology is more likely
to be adopted at the pattern where the associated transportation cost is lower. There
are three justifications. First, the range of tT for which the modern technology undercuts
the traditional one is wider at the pattern with the lower transportation cost, say pattern
k ∈ S, C. It means that the transportation sector faces a less stringent undercutting
condition at that pattern. Second, if the sector cannot provide the transportation service
profitably at pattern k, then nor can it do so at the other pattern. Conversely, if it can
provide the service profitably at the latter pattern, it can do so also at pattern k. Third
and finally, there is a possibility that the modern technology is adopted only at pattern k.
Yet there is no possibility that it is adopted only at the other pattern.
Unfortunately, however, it turns out that which of tS and tC is smaller is ambiguous. It
is true that they are negatively related to DS and DC , respectively (see (12)), and, therefore,
that the transportation cost is lower at the distribution pattern with a larger volume of inter-
regional trade. But the volume of trade at the symmetric pattern depends in turn on the
transportation cost (see (15) ). In this sense, the level of transportation cost and the volume
15
of inter-regional trade are inter-related. Thus, the answer to the question of which of tS and
tC is smaller is not obvious. In this section, I explore how the answer depends on the values
of parameters.
4.1 Effect of σ
I begin the analysis by examining the effect of σ. Let us return to Fig. 1, where the upward-
sloping part of the tS(σ) curve, denoted by tS(σ) curve hereafter, and the whole tC(σ) curve
represent tS and tC , respectively. Depending on how they intersect each other, we can
distinguish three cases. First, suppose that the tC(σ) curve lies above the tS(σ) curve at
σ = σ. This occurs if Θ(F, γ) > 0, where
Θ(F, γ) ≡ γ(σ − 1)σ − 1− σF
−[2− F
σF
]1/(σ−1)
.
Since the tC(σ) curve is downward-sloping, the tS(σ) curve lies below the tC(σ) curve in the
interval σ ∈ [1/(1− F ), σ], where 1/(1−F ) is a minimal value of σ permitted by Assumption
1. Therefore, there is no intersection. Second, suppose that the tC(σ) curve intersects the
tS(σ) curve at σ = σ. This occurs if Θ(F, γ) = 0. In this case, that intersection becomes a
unique intersection of the tS(σ) curve and the tC(σ) curve. Finally, suppose that the tC(σ)
curve lies below the tS(σ) curve at σ = σ as in Fig. 1. This occurs if Θ(F, γ) < 0. Since the
tC(σ) curve approaches the positive infinity as σ goes to 1/(1 − F ) from above, the tS(σ)
curve and the tC(σ) curve necessarily intersect each other. Furthermore, the intersection is
unique because the tC(σ) curve is downward-sloping. I refer to σ corresponding to such an
intersection as σ∗ with σ∗ ∈ [1/(1− F ), σ). It is evident that the tS(σ) curve lies above the
tC(σ) curve for σ greater than σ∗ whereas the opposite is true for σ smaller than σ∗. Thus,
we have established the following proposition.
Proposition 3. If Θ(F, γ) > 0, then tS < tC for any σ ∈ [1/(1− F ), σ]. Otherwise, there
exists σ∗ ∈ [1/(1− F ), σ] such that tS<=>
tC if σ<=>
σ∗ for σ ∈ [1/(1− F ), σ].
As σ declines, it becomes more likely that σ < σ∗ and, consequently, that tS < tC .
That is, the lower the elasticity of substitution is, the more likely it is that the modern
transportation technology is adopted at the symmetric pattern but not at the core-periphery
pattern. This is explained as follows. As the elasticity declines, the user fee declines at the
symmetric pattern but rises at the core-periphery pattern, which we have seen earlier by the
comparative statics analysis. Consequently, it becomes more likely that the transportation
cost at the former pattern falls short of the counterpart at the latter pattern.
16
4.2 Effect of F
Next, I examine the effect of F . In Fig. 2, the tS(F ) curve shows the measure of trans-
portation cost at the symmetric pattern (tS) as a function of F , given σ and γ. At its
turning point, namely, at F = F , maxtS
Ω(tS) = 0 and therefore (17) has a unique solution,
t0S . By construction, the increasing part of the curve gives tS , which is a convex function
of F . On the other hand, the tC(F ) curve shows the measure of transportation cost at the
core-periphery pattern (tC). It is increasing and a convex function of F . Both the tS(F ) and
tC(F ) curves cut the vertical axis at γ. Furthermore, vertical line F = ρ, which does not
appear in the figure, has a double significance. First, only the region at its left is relevant
since ρ is the upper bound of F that satisfies Assumption 1. Second, (22) implies that the
line is an asymptote of the tC(F ) curve.
@Insert Fig. 2 around here
As an illustration, consider the case where σ = 2 (ρ = 0.5). The first panel in Fig.
2 describes the case with γ = 5, for which F = 0.0911. Here, over the interval with
F ∈(0,min[F , ρ]
], the tS(F ) curve lies above the tC(F ) curve, that is, tS > tC . Now,
suppose that γ gradually declines from 5 with σ being kept at 2. Then, both the tS(F ) and
tC(F ) curves shift downward (recall that ∂tS/∂γ > 0 and that ∂tC/∂γ > 0 ). As soon as γ
drops to 3.258, the increasing part of the tS(F ) curve begins to intersect the tC(F ) curve at
positive F . Thus we have entered the second phase, which is represented by the case with
γ = 2 (see Fig. 2 (b)). Let us denote the value of F at the intersection by F ∗ ∈ (0, F ].
That is, tS<=>
tC , if F<=>
F ∗, respectively, provided that F ∈(0,min[F , ρ]
]. As γ falls
further, F ∗ rises, and eventually the two curves come to intersect each other at F = F
when γ = 1.286. Thereafter, the tS(F ) curve intersects the tC(F ) curve not at its increasing
part but at its decreasing part: tS < tC for F ∈(0,min[F , ρ]
]. This is the third phase,
represented by the case where γ = 1.1 (see Fig. 2 (c)). Let us define Ψ(σ, γ) and Φ(σ, γ) as
Ψ(σ, γ) ≡(1 + γσ−1
) (1 + σγσ−1
)(σ−1)2−8σ2 and Φ(σ, γ) ≡ F
[2σ2 − (σ − 1)2
]−2(σ−1),
respectively. The next proposition, whose proof is relegated to Appendix, states the above
findings in a more rigorous manner.
Proposition 4. i) If Ψ(σ, γ) > 0, then tS > tC for any F ∈(0,min[F , ρ]
].
ii) If Ψ(σ, γ) < 0 and Φ(σ, γ) ≤ 0, then there exists F ∗ ∈(0,min[F , ρ]
]such that
tS<=>
tC if F<=>
F ∗ for any F ∈(0,min[F , ρ]
]. (23)
iii) If Ψ(σ, γ) < 0 and Φ(σ, γ) > 0, then tS < tC for any F ∈(0,min[F , ρ]
].
Suppose that Ψ(σ, γ) < 0 and Φ(σ, γ) ≤ 0. Then, as F declines, it becomes more likely
that F < F ∗ and that tS < tC . In other words, it becomes more likely that the modern
17
transportation technology is adopted at the symmetric pattern rather than at the core-
periphery pattern. This is explained as follows. Suppose that the fixed cost declines. If the
user fee were kept constant, the profit of the transportation sector would increase. To break
even, the fee and, therefore, the transportation cost need to decline since the elasticity of
the cross-border demand is smaller than 1 (recall that ∂tS/∂F > 0 and that ∂tC/∂F > 0).
Now, we know that the elasticity is positive at the symmetric pattern while it is 0 at the
core-periphery pattern, that is to say, it is greater at the former pattern. Consequently, the
transportation cost needs to decline more sharply at the former pattern.
4.3 Effect of γ
By defining Ξ(σ, F ) ≡ ρ(σ − 1)(2 − F )KC − 2, we obtain the following result for the effect
of γ, whose proof is relegated to Appendix.
Proposition 5. i) If Ξ(σ, F ) < 0 and Ω (KC(σ − 1)) ≤ 0, then tS > tC for any γ ∈ (1, γ].
ii) If Ξ(σ, F ) < 0 and Ω (KC(σ − 1)) > 0, or if Ξ(σ, F ) = 0, then there exists γ∗ ∈ (1, γ]
such that
tS<=>
tC if γ<=>
γ∗ for any γ ∈ (1, γ]. (24)
iii) If Ξ(σ, F ) > 0, then tS < tC for any γ ∈ (1, γ].
Consider case ii). When γ is lower, it is more likely that tS < tC . In other words, the
more efficient the technology is, the more likely it is to be adopted at the symmetric pattern
but not at the core-periphery pattern. This is explained as follows. Suppose that we are
now provided with a more efficient modern technology. If the user fee were kept constant,
a 1 % decrease in γ would result in a 1% decrease in the measure of transportation cost
(tS or tC) no matter which distribution pattern prevails in the economy. At the symmetric
pattern, however, the lower transportation cost, being associated with a larger cross-border
demand, implies a higher revenue of the transportation sector. Consequently, the user fee
needs to decline to break even. Hence, a 1% decrease in γ actually brings about more than
1% decrease in tS . At the core-periphery pattern, to the contrary, the amount of the cross-
border demand does not depend on the transportation cost. Therefore, the revenue does
not change and the user fee remains constant. Thus, a 1% decrease in γ brings about merely
1% decrease in tC .
5 Stable Long-Run Equilibrium Pattern
In this section, I study the distribution pattern of entrepreneurs that emerges for a given
transportation technology. First of all, it must be a long-run equilibrium pattern at which
18
no entrepreneur has an incentive to migrate. Let V1 and V2 be the levels of indirect utility
for an entrepreneur in respective regions, and V ≡ V1/V2.10 Then, a distribution pattern is
a long-run equilibrium pattern when the following condition is satisfied:V = 1 if λ ∈ (0, 1)
V ≥ 1 if λ = 1
V ≤ 1 if λ = 0.
(25)
In other words, at the equilibrium pattern, the levels of the indirect utility must be equal in
the two regions when there are entrepreneurs in both regions. When they are concentrated
in one region, instead, they could not enjoy a higher level of utility even if they migrated to
the other region unilaterally.
Among the long-run equilibrium patterns, furthermore, I focus on stable ones. Suppose
that the economy is initially characterized by the long-run equilibrium pattern, and then a
small mass of entrepreneurs migrates from one region to the other. I say that the equilibrium
pattern is stable if the indirect utility in the region they arrive at becomes lower than that
in the region they has left. Thus, when V 6= 1, the equilibrium pattern is necessarily stable.
When V = 1, on the other hand, it is stable if dV /dλ < 0 and unstable if dV /dλ ≥ 0.11 To
sum up, we have the following definition.
Definition 2. (SLE pattern) For a given transportation technology i ∈ T,M, a dis-
tribution pattern k ∈ S, C is said to be a stable long-run equilibrium pattern or an SLE
pattern if
i) (25) is satisfied; and, in addition,
ii) either or both of V 6= 1 and dV /dλ < 0 holds.
The set of these conditions is referred to as an SLE condition.
5.1 SLE condition for the Symmetric Pattern
First, let us concentrate on the symmetric pattern. Since V = 1, it is always a long-run
equilibrium pattern. Furthermore, note that
dV
dλ= 4 · tσ−1 − 1
tσ−1 + 1· (σ − 1)(2− σ)tσ−1 + σ(σ + 1)
(σ − 1) [(σ − 1)tσ−1 + σ + 1].
10A tedious computation yields V =“t1−σ + Z
”/
hP
“1 + t1−σZ
”iwhere P ≡ P1/P2 and
Z ≡Z1
Z2=
(1− λ)(σ − 1)tσ−1 + λ(σ + 1)
λ(σ − 1)tσ−1 + (1− λ)(σ + 1).
11I consider that the distribution pattern is unstable when dV /dλ = 0. One could obtain the similar
results considering otherwise.
19
If σ ≤ 2, then dV /dλ > 0 and the pattern is unstable. To focus on an interesting case, let us
preclude this possibility, assuming that σ > 2, which is known as a no-black-hole condition
in the literature.
Assumption 2. (no-black-hole condition) σ > 2.
If the no-black-hole condition is satisfied, we have dV /dλ<=>
0 if t>=<
τS where τS ≡
[σ(σ + 1)/(σ − 1)(σ − 2)]1/(σ−1). By construction, the symmetric pattern is stable if and
only if t > τS . Here, one would notice that τS corresponds to the “break point” in Fujita,
Krugman and Venables (1999): as the transportation cost gradually declines from a suffi-
ciently high level, the symmetric pattern becomes unstable, that is, the symmetry breaks,
at a certain point, namely, the break point.
Proposition 6. Suppose that a transportation technology i ∈ T,M is adopted. Then, the
symmetric pattern is an SLE pattern if and only if t > τS is satisfied, where t = tT if i = T ,
and t = tS if i = M .
Now, it is useful to derive several properties of τS . First, it depends only on σ. Second,
it is a decreasing function of σ due to the no-black-hole condition. Finally, it approaches 1
as σ goes to the positive infinity, and approaches the positive infinity as σ goes to 2 from
above. For the case where γ = 1.1 and F = 0.1, τS is depicted in Fig. 1 as a τS(σ) curve.
Proposition 6 states that the symmetric pattern is an SLE pattern if parameters lie above
the τS(σ) curve.
5.2 SLE condition for the Core-Periphery Pattern
Next, let us turn to the core-periphery pattern. Without loss of generality, I analyze the
case where all the entrepreneurs are concentrated in region 1, that is, λ = 1.
Since V = 2σtσ/[(σ − 1)t2(σ−1) + σ + 1
], we have V
>=<
1, as Γ(σ, t)>=<
0, where
Γ(σ, t) ≡ 2σtσ − (σ − 1)t2(σ−1) − (σ + 1).
Several observations follow with respect to Γ(σ, t). First, Γ(σ, 1) = 0 for any σ. Second,
since ∂Γ(σ, t)/∂t = 2tσ−1[σ2 − (σ − 1)2tσ−2
], we have ∂Γ(σ, t)/∂t
>=<
0, if t<=>
t for any
σ > 2, where t ≡ [σ/(σ − 1)]2/(σ−2). Finally, limt→∞
Γ(σ, t) < 0 for any σ > 2. From those
observations, we can figure out how Γ(σ, t) changes with t for a given value of σ. As t rises
from 1, Γ(σ, t) at first grows from 0 until t reaches t, then declines, becomes equal to 0 when
t hits a certain value, τC > t, and finally becomes negative. Therefore, we have
Γ(σ, t)>=<
0 if t<=>
τC (26)
20
for a given σ > 2 (remember that t > 1). Thus, if t < τC , we have V > 1: the core-
periphery pattern is an SLE pattern. If t = τC , it is a long-run equilibrium pattern since
V = 1; but is unstable because dV /dλ = 0. Finally, if t > τC , it is not a long-run equilibrium
pattern. One may notice that τC corresponds to the “sustain point” in Fujita, Krugman
and Venables (1999): as the transportation cost gradually decline from a sufficiently high
level, the core-periphery pattern becomes an SLE pattern, that is, it becomes sustainable
at a certain point, namely, at the sustain point.
I have established the following proposition.
Proposition 7. Suppose that a transportation technology i ∈ T,M is adopted. Then, the
core-periphery pattern is an SLE pattern if and only if t < τC is satisfied, where t = tT if
i = T , and t = tS if i = M .
Now, it is useful to derive several properties of τC . First, it depends only on σ. This
is because Γ(σ, t) does not depend on the parameters other than σ and t. Second, it is
a decreasing function of σ.12 Third and finally, it approaches 1 as σ goes to the positive
infinity, and approaches the positive infinity as σ goes to 2 from above.13 Notice that all of
these properties are similar to the counterparts of τS , though their derivation is much more
complicated for τC . For the case where γ = 1.1 and F = 0.1, τC is depicted in Fig. 1 as
a τC(σ) curve. The core-periphery pattern is an SLE pattern if parameters lie below the
τC(σ) curve.
5.3 SLE conditions and Distribution Patterns: A Comparison
One important result concerns the relative sizes of τS to τC . By a tedious computation, I
derive the following proposition, whose proof is relegated to Appendix.
Proposition 8. τC > τS for a given value of σ > 2.
This indicates that the τC(σ) curve lies above the τS(σ) curve in Fig. 1. As should be clear
from the preceding discussion, furthermore, the proposition means that the “sustain point”
exceeds the “break point”. This result is the same as the one shown through a numerical
simulation by Fujita, Krugman and Venables (1999) and shown analytically by Forslid and
Ottaviano (2003).
The following corollary, whose proof is relegated to Appendix, immediately follows:12Differentiating Γ (σ, τC) = 0 yields
dτC
dσ= −
E + 2τσCG ln τC
2(σ − 1)2τσ−1C H
.
Here, E ≡ 1− 2τσC + τ
2(σ−1)C =
hτ2(σ−1)C − 1
i/σ > 0 where (26) is used. Furthermore, τC > t implies that
G ≡ (σ − 1)τσ−2C − σ > 0 and that H ≡ τσ−2
C − [σ/(σ − 1)]2 > 0. Therefore, the differential is negative.13Since lim
σ→2t = ∞ and τC > t, we must have lim
σ→2τC = ∞.
21
Corollary 1. Suppose that the traditional technology is adopted. If k ∈ S, C is not an
SLE pattern, then l 6= k (l ∈ S, C) is an SLE pattern.
6 Interdependence of Transportation Technology and
Economic Geography
So far, we have discussed the adoption of transportation technology given the economic
geography in Sections 3 and 4, and, subsequently, the SLE distribution pattern given the
transportation technology in Section 5. The remaining task is the simultaneous determina-
tion of the transportation technology and the economic geography : what combination of the
technology and the geography is realized in the economy?
6.1 Maintainability
Yet, first of all, when should we consider a particular combination to be “realized”? There
would be a variety of answers with different levels of strictness in the criteria; but, in this
paper, I adopt a simple one. For one thing, the transportation technology must be indeed
adopted in the economy in the light of Definition 1. Otherwise, either the transportation
sector, if any, would suffer a loss (the modern technology would not be available to con-
sumers) or consumers would attempt to use the other technology (the relevant undercutting
condition would not be satisfied). Next, the distribution pattern must be an SLE pattern.
Otherwise, the pattern would be going to change. That is, if it were not a long-run equi-
librium pattern, further adjustment would occur in the migration process of entrepreneurs.
If it were not stable, on the other hand, a small perturbation would provoke an influx of
migration. A set of these two criteria gives the following definition of maintainability.
Definition 3. (maintainability) A pair (i, k) is said to be maintainable if the trans-
portation technology i ∈ T,M is adopted in the economy given the distribution pattern
k ∈ S, C and, at the same time, the distribution pattern k ∈ S, C is an SLE pattern
given the transportation technology i ∈ T,M.
Using the results obtained so far, we can find the range of parameters for which each of
the four possible pairs, (T, S), (M,S), (T,C) and (M,C), is maintainable. Fig. 3 depicts
the same situation as Fig. 1 (γ = 1.1 and F = 0.1) does. The sets of (σ, tT )’s that make
pairs (T, S), (M,S), (T,C) and (M,C) maintainable, respectively, are represented by the
shaded areas in Figs. 3. (a)-(d).
@Insert Fig. 3 around here
22
It is worth noting that the concept of maintainability is exclusive in terms of the trans-
portation technology. That is to say, if pair (i, k) is maintainable, then pair (j, k) with j 6= i
is not maintainable: whenever a particular technology constitutes a maintainable pair, the
other technology does not do so as long as the distribution pattern is the same. This result
follows directly from the definition of the adoption of technology. On the contrary, the con-
cept of maintainability is not necessarily exclusive in terms of the distribution pattern: that
pair (i, k) is maintainable does not preclude the possibility that pair (i, l) with l 6= k is also
maintainable. This can be easily verified from the fact that the shaded areas in (a) and (b)
(and also those in (c) and (d)) of Fig. 3 have a non-empty intersection. Thus, the same
technology can constitute a maintainable pair with each of the two distribution patterns.
6.2 Lock-in Effect
Paying attention to the interdependence of transportation technology and economic geog-
raphy enables us to discuss the possibility of a lock-in effect. It concerns the situation in
which a transportation technology fails to be adopted given a particular distribution pattern
despite the fact that it could be adopted if the distribution pattern were different. More
specifically, this is a situation in which both pair (T, k) and pair (M, l) with l 6= k are
maintainable. Here, if the realized pattern happens to be k, the economy is locked in to the
inferior state where the traditional technology is adopted: if the realized pattern is not k
but l, it succeeds in adopting the modern technology and is locked in to the superior state.
The following statement gives a formal definition:
Definition 4. (lock-in effect) The economy is locked in if both pair (i, k) and pair (j, l)
are maintainable for i ∈ T,M, k ∈ S, C, j ∈ T,M with j 6= i, and l ∈ S, C with
l 6= k.
It is worth while studying the lock-in effect not only because it is theoretically significant
but also because it has some important implications and applications.
First, the lock-in effect justifies a policy intervention. If the inferior state with the tra-
ditional technology is realized once, the economy cannot attain the superior state with the
modern technology by a decentralized mechanism when the lock-in effect is present. To es-
cape from this poverty trap, consequently, some sort of policy intervention is necessary. The
government may, for example, take a certain measure stimulating entrepreneurs to change
their locations so that the pair associated with the inferior state becomes not maintainable.
Second, the lock-in effect provides us with one explanation for a puzzling fact that some
countries succeed in adopting the modern technology while the others cannot, even if their
economic environments are not much different. Our explanation is that there is no intrinsic
23
divergence between the two groups of countries: some are luckier than the others. Here,
usual argument concerning the multiple equilibria applies. That is, we are attempting to
reveal what combination of the transportation technology and the economic geography can
be realized in the economy but not what combination is actually realized. The latter question
would be answered in terms of the factors which are out of consideration in this paper, such
as historical accidents.
An immediate question is which distribution pattern is associated with the traditional
technology. It may be a symmetric pattern when pair (T, S) and pair (M,C) are main-
tainable. Or, it may be a core-periphery pattern when pair (T,C) and pair (M,S) are
maintainable. Mobilizing the requirements for the maintainability, we can easily establish
the following proposition, which is thus presented without a proof.
Proposition 9. i) The economy is locked in if all of the following conditions, a)-d), are
satisfied:
a) either σ > σ or tT < tS,
b) tT > τS,
c) tT ≥ tC , and
d) tC < τC .
In this case, the symmetric pattern is associated with the traditional technology while the
core-periphery pattern is associated with the modern technology.
ii) The economy is locked in if all of the following conditions, a)-d), are satisfied:
a) σ ≤ σ,
b) tS ≤ tT < tC ,
c) tS > τS, and
d) tT < τC .
In this case, the core-periphery pattern is associated with the traditional technology while the
symmetric pattern is associated with the modern technology.
In Fig. 4, which is drawn for the same values of parameters as Fig. 1 ( γ = 1.1 and
F = 0.1), the shaded area represents the set of (σ, tT )’s for which the five conditions in i)
are simultaneously satisfied. Obviously, it is the intersection of the set for which pair (T, S)
is maintainable and that for which pair (M,C) is maintainable.
@Insert Fig. 4 around here
The other case, described in ii) of Proposition 9, is not plausible to occur. When γ = 1.1
and F = 0.1, there is no (σ, tT )’s for which the four conditions in ii) are satisfied at the same
time. Then, one might guess that the same applies for any γ and F . Since the conditions
are highly nonlinear and some of them take an implicit form, however, it is not possible to
24
prove it analytically. By numerical simulation, we can show that the case described in ii) is
unlikely to occur.14
To sum up, there is a possibility that the economy is locked in to the inferior state with
the traditional technology when it is characterized by the symmetric pattern. When it is
characterized by the core-periphery pattern, instead, it is unlikely that the economy is locked
in to the inferior state.
6.3 An Application: Economic Development and the Endogenous
Change of Transportation Technology
Applying the concept of maintainable pair, we can examine how the adopted transportation
technology and the emerging economic geography change as a result of an exogenous shock
in an economic environment. In this subsection, I examine the change brought by a gradual
decline in F , the fixed cost measured by the final good, as an illustration. The decline
can be regarded as one of the consequences of economic development when one takes a
sufficiently long time horizon.15 It would be also possible to analyze the decline in γ, the
inverse measure of the efficiency of the modern transportation technology, which might be
another consequence of economic development. It is, however, omitted due to the limitation
of space.
I assume that all the adjustments, not only the short-run adjustment in prices, consump-
tion and production but also the long-run adjustment in entrepreneurs’ location take place
rapidly compared to the speed of the decline in F . That is to say, we consider the following
world of artifact. At the beginning of each time period, F declines by a small amount.
Then, F remains constant up until the end of that period. All the while, the adjustments
are made and a maintainable pair is realized. Then, at the beginning of the next period, F
declines again. It remains constant for a while and the adjustments follow; and the process
continues. Of course, for a more rigorous and throughout treatment, it would be necessary
to formulate the dynamic process explicitly. However, it would demand sizable length of14In order for the four conditions to be satisfied, it is necessary that the value of σ at which the increasing
part of the tS(σ) curve (tS(σ) curve ) intersects the τS(σ) curve is lower than σ (see Fig. 4). If this is not
met, σ ≤ σ and σ > τS cannot be satisfied at the same time, since the τS(σ) curve is downward-sloping.
Another necessary condition is that the tS(σ) curve intersects the tC(σ) curve at a greater value of σ than
the former intersects the τS(σ) curve. If this is violated as in Fig. 4, tS ≤ tT < tC and tS > τS cannot
hold simultaneously, because both the tC(σ) curve and the τS(σ) curve are downward-sloping. A simulation
analysis suggests that no relevant combination of parameters satisfies these two necessary conditions at the
same time.15In the literature of economic geography, it is not rare that economic development is expressed, explicitly
or implicitly, by the gradual change in an exogenous parameter, noticeably, the expansion of population or
the decline in transportation cost. The examples abound in Fujita, Krugman and Venables (1999).
25
analysis without yielding much additional insight. Therefore, I rather take a more or less
intuitive approach, sacrificing formality.
In Fig. 5, tS(F ) and tC(F ) curves describe tS and tC , respectively, as functions of F as
in Fig. 2, for σ = 4.5 and γ = 1.1. In addition, τS and τC , which are now constant, are
shown by horizontal lines.16 Now, I pick up three representative cases and, with a help of
the figure, describe the trajectory along which the economy evolves over time.
In the first case, the transportation cost associated with the traditional technology is
relatively high. In Fig. 5, the initial values of F and tT are represented by point A. Because
(T, S) is a unique maintainable pair for those parameter values, it is natural for us to predict
that (T, S) is realized in the economy.
@Insert Fig. 5 around here
As the economy develops and F declines, we move along the horizontal line passing
through point A. As soon as the economy reaches point B where tC = τC , the core-periphery
pattern becomes an SLE pattern given the modern technology. Then, both (T, S) and
(M,C) become a maintainable pair. Here, we encounter the problem of the multiplicity of
maintainable pair. To avoid an unnecessary complication, I take a rather ad hoc approach
of law of inertia, which says that the economy moves toward the state with the smallest
friction. If the economy shifts from the state with (T, S) to the state with (M,C), both
the transportation technology and the distribution pattern need to change. However, it can
remain at the state with (T, S) involving no change in the technology nor in the pattern. In
this sense, the “friction” incurred in the latter scenario is smaller than that in the former.
Therefore, I consider that the economy sticks to (T, S).
The economy enters a different state when F hits F at point C. For this value of F ,
the modern technology becomes available and undercuts the traditional technology given
the symmetric distribution. Therefore, (M,S) becomes a maintainable pair whereas (T, S)
ceases to be so. Now, we have two maintainable pairs, (M,S) and (M,C). Applying the
law of inertia, one can conclude that (M,S) will be realized in the economy.17 Here, the
economy jumps to point D; and the transportation cost decreases by a discrete amount, that
is, the change is discontinuous. Afterward, the transportation cost decreases continuously
as tS moves along the tS(F ) curve.
The next turning point comes at point E. As soon as the economy reaches it, the sym-
metric pattern becomes unstable and ceases to be an SLE pattern or “breaks”, because tS
16The parameter values give F = 0.104, t0S |F=F = 1.491, τS = 1.346 and τC = 1.368.17The shift from the state with (T, S) to that with (M, S) involves a change only in the technology while
the shift to the state with (M, C) involves changes in both the technology and distribution pattern. Thus,
the former shift incurs smaller friction than the latter.
26
is no longer higher than τS . Therefore, (M,S) ceases to be a maintainable pair. Notice that
(M,C) has already been a maintainable pair. It now becomes a unique maintainable pair.
Thus, we can predict that the state changes from that with (M,S) to that with (M,C).
Recall that the adjustment in the entrepreneurs’ location is assumed to be sufficiently rapid
compared to the change in F . As a limiting case, we might consider that the adjustment
is made immediately. Then, the economy jumps from point E to point F with a discrete
decrease in the transportation cost. After this change, it evolves along with the tC(F ) curve.
Here, the transportation cost gradually declines.
Next, let us turn to the second case where the transportation cost associated with the
traditional technology is of moderate size. The initial values of F and tT are represented by
point G in Fig. 5. For those values, we have two maintainable pairs, (T, S) and (T,C). Our
framework cannot prescribe from which pair the economy starts.
On the one hand, suppose that (T, S) is initially chosen. As F declines, it moves along
the horizontal line passing through point G. Suppose that the economy reaches at point H.
If it were characterized by the core-periphery pattern, the modern technology, which would
have been available, now could undercut the traditional technology: (M,C) becomes a
maintainable pair and (T,C) ceases to be so. Nonetheless, the modern technology is still
not available given the symmetric pattern. Therefore, (T, S) continues to be a maintainable
pair. Here, we end up with the two maintainable pairs, (M,C) and (T, S). This is the
situation with the lock-in effect. By the law of inertia, the latter remains to be realized
in the economy. After it comes to point I where tT = tS , the traditional technology is
undercut by the modern technology, which has been available, given the symmetric pattern.
Therefore, the state will switch to (M,S). Then, the economy traces the same path as in
the previous case.
On the other hand, suppose that (T,C) is initially realized. As has been discussed,
the traditional technology is undercut by the modern technology, given the core-periphery
pattern, after point H. Therefore, the economy shifts to the state with (M,C) and enjoys
the superior modern technology, whereas, in the previous scenario, it sticks to the state with
(T, S). This is an example of how history matters when the lock-in effect is present. Finally,
after point H, there is no change in the state; and the transportation cost continuously
declines along with the tC(F ) curve.
The third and the final case, in which the economy starts from point J, is not too
interesting: there is always a unique maintainable pair; and the modern technology replaces
the traditional one as soon as the former comes to undercut the latter.
Several observations follow. First, our framework can provide a complete explanation for
why and how the adopted transportation technology and the emerging economic geography
27
change as a result of the decline in F . In the explanation, their interdependence plays a
critical role. Second, a small difference in the initial values of parameters may result in
a big difference in the paths of the transportation cost. This is because it can provoke
the divergence in the adopted technology and the realized geography. Third, even with
the same parameter values, an economy may experience different paths depending on the
initial geography, as is described by the second case. This occurs when there exist multiple
maintainable pairs. Finally, our framework gives a story often uttered in the standard
literature, the story that the symmetric distribution breaks and then the core-periphery
pattern emerges as a result of economic development.
7 Concluding Remarks
In this paper, I have studied the interdependence of economic geography and transportation
technology. Constructing a simple two-region model, I have examined the conditions for the
modern transportation technology to be adopted in an economy. In particular, the impact
of economic geography upon the adoption of the modern technology has been explored.
Furthermore, I have discussed which combination of the economic geography (symmetric or
core-periphery pattern) and the transportation technology (traditional or modern technol-
ogy) is to be realized in an economy. Among others, the possibility that it is locked in to
an inferior state has been shown. When it occurs, the economy fails to adopt the modern
technology at a given economic geography, although it could successfully adopt the technol-
ogy if the economic geography were different. Finally, I have examined the changes in the
adopted transportation technology and the realized economic geography due to a gradual
change in an exogenous factor, which is associated with economic development.
It should be noted nevertheless that this study is only a first step toward the throughout
understanding of the endogenous determination of transportation technology. Among the
limitations of this paper, the followings seem to be especially important. First, I have relied
on a two-region model, which allows us to take into account only the symmetric and the
core-periphery patterns as a distribution of economic activities. In the reality, however, there
are more than two regions, or it may be even better to consider that economic activities
are distributed over a continuous space. It would be necessary for us to recognize a richer
variety of economic geography. Second, we have considered only one class of the modern
transportation technology. By admitting more than one class, one would be able to discuss
some interesting problems such as the selection between a “heavy” transportation technology
with higher fixed cost plus lower operation cost, and a “light” technology with lower fixed
cost plus higher operation cost. Third and finally, we have focused upon the special situation
28
in which the transportation sector earns no profit. In recent years, however, it has been
becoming more and more common in many developed countries that the transportation
sector, in particular, railways, is run by private companies. It is plausible that its profit-
seeking behavior alters our results drastically.
29
References
[1] Behrens, Kristian (2004). “On the Location and ‘Lock-In’ of Cities: Geography vs.Transportation Technology,” mimeo.
[2] Forslid, Rikard, and Gianmarco I. P. Ottaviano (2003). “An Analytically SolvableCore-Periphery Model,” Journal of Economic Geography, 3, 229-40.
[3] Fujita, Masahisa, Paul Krugman, and Anthony J. Venables (1999). The Spatial Econ-omy: Cities, Regions, and International Trade, The MIT Press, Cambridge, Mass.
[4] Fujita, Masahisa, and Jacques-Francois Thisse (2000). “The Formation of EconomicAgglomerations: Old Problems and New Perspectives,” in Economics of Cities, eds. byJ.-M. Huriot and J.-F. Thisse, Cambridge University Press, Cambridge, UK.
[5] Fujita, Masahisa, and Jacques-Francois Thisse (2002). Economics of Agglomeration,Cambridge University Press, Cambridge, UK.
[6] Krugman, Paul (1991). “Increasing Returns and Economic Geography,” Journal ofPolitical Economy, 99, 483-99.
[7] Mori, Tomoya and Koji Nishikimi (2002). “Economies of Transport Density andIndustrial Agglomeration,” Regional Science and Urban Economics, 32, 167-200.
[8] Mori, Tomoya and Alessandro Turrini (2005). “Skills, Agglomeration, and Segmenta-tion,” European Economic Review, 49, 201-25.
[9] Murphy, Kevin M., Andrei Shleifer, and Robert W. Vishny (1989). “Industrializationand the Big Push,” Journal of Political Economy, 97, 1003-26.
[10] Neary, Peter J. (2001). “Of Hype and Hyperbolas: Introducing the New EconomicGeography,” Journal of Economic Literature, 39, 536-61.
[11] Ottaviano, Gianmarco, and Diego Puga (1998). “Agglomeration in the Global Economy:A Survey of the ‘New Economic Geography’,” World Economy, 21, 707-31.
30
Appendix
In some of the subsequent proofs, common notations are used: D0 ≡ tS − tC , D1x ≡
∂tS/∂x− ∂tC/∂x and D2x ≡ ∂2tS/∂x2 − ∂2tC/∂x2 for x ∈ σ, F, γ.
Proof of Lemma 1.If the modern technology is available and feasible, and undercuts the traditional tech-
nology, ii) of Definition 1 indicates that the modern technology is adopted. (Note that thetraditional technology is always available and feasible.) Next, if the modern technology isunavailable and/or infeasible, it follows from i) of Definition 1 that the traditional tech-nology is adopted. Finally, if the modern technology is available and feasible but does notundercut the traditional technology, ii) of Definition 1 implies that the traditional technol-ogy is adopted. QED
Proof of Proposition 4.At F = F , on the one hand, the tS(F ) curve becomes a vertical line while the tC(F )
curve has a positive slope provided that F < ρ. For F < F , on the other hand, tS < t0Sby construction. It implies that tS < 2σγ/(σ − 1)(2− F ) since (17) prescribes that tσ−1
S =[(2− F )tS − 2γ] /(FtS). Therefore, we have
D1F |tS=tC
=(KC)2σ(σ − 1)(σ + 1)FγtC
2σγ − (σ − 1)(2− F )tS> 0
for any F > 0: the increasing part of the tS(F ) curve (tS(F ) curve) is steeper than thetC(F ) curve whenever they pass through the same point. Hence, it is not possible for thetwo curves to be tangent to each other at any F ∈
(0,min[F , ρ]
]; and, furthermore, whenever
they intersect within that interval, the tS(F ) curve cuts the tC(F ) curve from below. I referto this as a relative steepness property of the two curves. Now, pay our attention to theneighborhood of F = 0. We have D0|F=0 = 0 and D1
F |F=0 = 0. Moreover, a tediouscomputation yields D2
F |F=0 = γΦ(σ, γ)/2(σ − 1)2 since tS = tC = γ at F = 0 and (17)implies that (tS −γ)/F = (tS + tσS)/2. I consider two cases. First, suppose that Φ(σ, γ) > 0.Then, we have D2
F |F=0 > 0. Therefore, tS > tC in the neighborhood of F = 0 with F > 0.Because of the relative steepness property, however, the two curves cannot intersect norcannot be tangent to each other. Consequently, the tS(F ) curve lies above the tC(F ) curvefor any F ∈
(0,min[F , ρ]
]. This establishes i). Second, suppose that Φ(σ, γ) < 0. In this
case, tS < tC in the neighborhood of F = 0 with F > 0. Furthermore, using (19) and thefact that tS |F=F =
[(2− F )/σF
]1/(σ−1), we have
D0|F=F =2γ
ρ(2− F )− (σ − 1)γ
σ − 1− σF= −γ
σΨ(σ, γ).
If Ψ(σ, γ) ≤ 0, on the one hand, tS ≥ tC at F = F . By the relative steepness property,therefore, the tS(F ) curve intersects the tC(F ) curve at its increasing part, and only once.Hence, there exists F ∗ ∈
(0, F
]that satisfies (23). Since Ψ(σ, γ) ≤ 0 implies that F < ρ,
such F ∗ lies in the interval(0,min[F , ρ]
]. Thus we have proved ii). If Ψ(σ, γ) > 0, on the
other hand, tS < tC at F = F . Because of the relative steepness property, it must be truethat tS < tC for any F ∈
(0, F
]and consequently, for F ∈
(0,min[F , ρ]
]. This establishes
iii). QED
31
Proof of Proposition 5.In parallel with the tS(F ) and tC(F ) curves, we can depict tS(γ) and tC(γ) curves that
give tS and tC , respectively, as a function of γ given σ and F . The tS(γ) curve turns aroundat γ = γ, where max
tS
Ω(tS) = 0 and, therefore, tS = t0S . Furthermore, its increasing part
(tS(γ) curve) gives tS . Because
D1γ |tS=tC
=(σ − 1)tC
[(2− F )t2Cγ
]γ [2σγ − (σ − 1)(2− F )tC ]
> 0
for any γ < γ, moreover, the tS(γ) and tC(γ) curves exhibit the relative steepness propertythat is similar to the counterpart the tS(F ) and tC(F ) curves exhibit (see Proof of Propo-sition 4). First, suppose that Θ(σ, F ) < 0, which implies that tC < tS at γ = γ, that is,tC |γ=γ < t0S . Since the tC(γ) curve is upward sloping, we have tC |γ=1 = KC(σ − 1) <
tC |γ=γ < t0S . Two cases are distinguished. On the one hand, suppose that Ω (KC(σ − 1)) =Ω (tC |γ=1) ≤ 0. In Section 3.1, I have argued that Ω(tS) > 0 for any tS ∈ (tS , tS). Therefore,it must be true that either tC ≤ tS or tC ≥ tS at γ = 1. Because tS ≥ t0S , the latter contra-dicts the fact that tC |γ=1 < t0S and, consequently, we have tC ≤ tS at γ = 1. The fact thattC < tS at γ = γ and tC ≤ tS at γ = 1, together with the continuity and the relative steep-ness property, implies that the tS(γ) curve lies above the tC(γ) curve for any γ ∈ (1, γ]. Thuswe have established i). On the other hand, suppose that Ω (KC(σ − 1)) = Ω (tC |γ=1) > 0,which implies that tC > tS at γ = 1. Since tC < tS at γ = γ, the two curves intersectwith each other, and only once, at some γ ∈ (1, γ]. Therefore, there exists γ∗ ∈ (1, γ] thatsatisfies (24). Second, suppose that Θ(σ, F ) = 0, that is, tC < tS at γ = γ. The relativesteepness property implies that tC < tS for any γ ∈ (1, γ). Consequently, (24) still holds ifwe set γ∗ = γ. This completes the proof of ii). Third and finally, suppose that Θ(σ, F ) > 0,that is, tC > tS at γ = γ. Due to the relative steepness property, the tS(γ) curve cannotintersect the tC(γ) curve at its increasing part. Thus, we have iii). QED
Proof of Proposition 8.18
According to (26), it is sufficient to show that Γ (σ, τS) > 0 for σ > 2. Note thatΓ (σ, τS) = 2σ2(σ + 1) (τS −K) /(σ − 1)(σ − 2), where K ≡ (σ3 − 2σ2 + 4σ2)/σ2(σ − 2).Taking a logarithm, we can obtain ln τS − lnK = ∆(σ)/(σ − 1), where
∆(σ) ≡ (2σ−1) ln σ− ln(σ−1)+(σ−2) ln(σ−2)+ ln(σ +1)− (σ−1) ln(σ3−2σ2 +4σ−2).
Three properties of function ∆(σ) are important. First, limσ→∞
∆(σ) = 0. Second,
limσ→∞
d∆(σ)/dσ = 0. Third and finally, d2∆(σ)/dσ2 > 0 for σ > 2. The last two properties
imply that ∆(σ) is a decreasing function. Together with the first property, consequently, wecan say that ∆(σ) > 0, which implies that Γ (σ, τS) > 0 by definition. QED
Proof of Corollary 1.First, suppose that the symmetric pattern is an SLE pattern. According to Proposition
6, tT ≤ τS . Since τS < τC by Proposition 8, it must be true that tT < τC , which implies thatthe core-periphery pattern is an SLE pattern (see Proposition 7). Next, suppose that thecore-periphery pattern is not an SLE pattern. Proposition 7 implies that tT ≥ τC . However,by Proposition 8, it implies that tT > τS . Hence, the symmetric pattern is an SLE patternby Proposition 6. QED
18The idea of this proof is based on a comment made by Takatoshi Tabuchi, which is greatly appreciated.
32